Cartan–Ambrose–Hicks theorem

In mathematics, the Cartan–Ambrose–Hicks theorem is a theorem of Riemannian geometry, according to which the Riemannian metric is locally determined by the Riemann curvature tensor, or in other words, behavior of the curvature tensor under parallel translation determines the metric.

The theorem is named after Élie Cartan, Warren Ambrose, and his PhD student Noel Hicks. Cartan proved the local version. Ambrose proved a global version that allows for isometries between general Riemannian manifolds with varying curvature, in 1956. This was further generalized by Hicks to general manifolds with affine connections in their tangent bundles, in 1959.

A statement and proof of the theorem can be found in